According to
this Reddit comment, that "battery" is a supercapacitor for keeping time (similar to the Xbox's RTC capacitor - except this one doesn't blow up).
Unfortunately the part they linked to (
EECEN0F204AK) is for the PSP fat motherboards (you can tell because the prongs stick out and are long) and no longer being made.
Looked around on Digikey for a similar looking part on the PSP 3000 motherboard and found the
EEC-EN0F204J2. It has a similar (if not exact) tab design ("horns" and curves down with a window), has 3 dots on top, has the M and + symbol, same capacitance (200 mF) and voltage (3.3v), same ESR rating (200mOhm), and they're both made by Panasonic.
Unfortunately again, the new part is discontinued. And the substitute part for the fat motherboards is completely out of stock.
If you feel like being the guinea pig, go ahead and use one of the regular battery coin cells. However I can't guarantee it'll end well (could blow up?)
EDIT: Looking a bit more, I found a
potential substitute part. The nice thing is it has higher capacitance than the regular one, but I don't know if this will cause an issue with the charge circuit (for the supercapacitor, not the battery). The main problem is it sits too high / takes up too much space.
